Parameter estimation procedure based on input-output integro-differential polynomials. Application to the Hindmarsh-Rose model

This paper deals with a parameter estimation method based on input-output integro-differential polynomials. From the Rosenfeld-Groebner algorithm, some differential relations depending only on the inputs, the outputs and the parameters of the model are obtained. A pretreatment consisting in some integrations of these relations permits to obtain new ones. The latter contain essentially integrals depending only on the model inputs/outputs and their higher order derivatives are lower than the initial relations. Therefore, they are less sensitive to the noise on the measurements compared to the initial ones. Integrating permits also to attenuate the effect of the noise improving by the same the parameter estimates. However, even if the numerical estimation algorithm provides a very good value of the parameters, the latter can lead to an incorrect behavior of the model output. Indeed, in biological or physical applications, a little change of some parameter values can lead to a radical change of the model behavior as for the Hopf bifurcation. A Hopf bifurcation refers to a radical change of the model output dynamic due to a parameter crossing a reference value. Therefore, an algorithm is proposed in this paper to test the reliability of any parameter estimation procedure with respect to the dynamic of the system. More precisely, from a given noise on the output(s), it consists in calculating the probability that the result of a parameter estimation algorithm will permit to reproduce the correct behavior of the model output. Finally, this algorithm is applied on the estimation procedure based on the input-output integro-differential polynomials and on the Hindmarsh-Rose model, a slow-fast model able to reproduce the main behaviors of a neuron and presenting a Hopf bifurcation.